The Ph.D. is designed to produce a skilled, professional mathematician who is trained to conduct research in mathematics, function effectively as a classroom teacher at the college level, or become a professional practitioner in an industrial or national laboratory setting.

Each candidate for the Ph.D. degree is required to complete a minimum of 60 hours of course work beyond the baccalaureate degree, including 12 credit hours of dissertation research and writing (MATH 899). Students are advised by a doctoral committee. This committee is generally chaired by the major professor (dissertation supervisor) and consists of at least five members, one from outside the department. The core members are writers of the student's Comprehensive Exams. A total of three credit hours of the variable credit doctoral seminar MATH 890 are required; these need not all be taken at once, rather credit is determined by the extent and intensity of participation. Students may earn these doctoral seminar credits by presentation of contemporary research in a student/faculty seminar in their research area.

Students pursuing the Ph.D. degree in mathematics are required to take three examinations: the Admission to Candidacy, Comprehensive, and Doctoral Defense Examinations.

The Admission to Candidacy Examination in mathematics consists of two three-hour written examinations and is administered with two options. The first examination for both options is based primarily, but not exclusively, on the content of the one-year sequence in real and complex analysis (MATH 703-704). The second examination for the first option is based primarily, but not exclusively, on the subject matter of the one-year sequence in abstract algebra (MATH 701-702). The second examination for the second option is based primarily, but not exclusively, on the subject matter of the one-year sequence in the foundations of computational mathematics (MATH 708-709). Two attempts of the Admission to Candidacy Examination are allowed. The first attempt should occur after the first year of graduate study and within the first two years of graduate study. The second attempt must be made at the next scheduled examination. Exceptions to the time constraint for unusual cases may be petitioned to the Graduate Director. Note that the exams are based upon the content of the various courses; it is not required that the well-prepared student take all, or even any, of these courses, although it is generally advisable to do so. Students need only be admitted candidacy once: if a student passes the exam based upon one of the options, say MATH 708-709 (or respectively MATH 701-702); but later wishes to specialize in an area for which the other option is more appropriate, then the content of MATH 701-702 (or respectively MATH 708-709) should be learned either by taking these courses or by independent study.

The Ph.D. Comprehensive is an in-depth examination consisting of a written part administered in three, three-to-four hour sessions, and an oral component. The written portion of the examination must either include the subject matter of one-year sequences numbered 710 or higher selected from two (or, exceptionally, three) of the eight areas listed in the Graduate Handbook, or, for the Concentration in ACM, from Groups 1 and 2 as described in the Graduate Handbook. See also the webpages for the Doctoral Program or the ACM Curriculum, respectively. In both cases, the subject matter of the student's research area should be tested in depth. The oral portion of the comprehensive will be based on the student's program of study and may include topics not covered by either the Admission to Candidacy Examination or the written portion of the Comprehensive Examination.

The Comprehensive Examination may be repeated only once. All portions of the examination must be completed within three weeks. As a general rule, the exam is offered twice each year, once in August and again in January, and should be taken after candidates have completed all or most of the courses required in their program, and before commencement of dissertation research. The examination must be completed at least 60 days prior to the receipt of the degree.

To complete the program, the student must write a dissertation (to be bound and delivered to the department), under the direction of a member of the graduate faculty, and defend the content of the dissertation in a final examination before the doctoral committee. It is expected that the content of the student's dissertation will be a significant contribution to the body of current research and will be published in a reputable journal.

To ensure breadth of mathematical training, each student is required to satisfactorily complete (B or better) 12 credit hours of course work in subject areas not covered by the Comprehensive Examination. Directed reading courses (MATH 798) may not be used to satisfy this requirement. Particular courses may be stipulated by the student's doctoral committee. The selection of the courses is subject to approval by the Graduate Director.